Hilbert symbol in the corresponding completion Qv. As usual, if v is the valuation attached to a prime number p then the corresponding completion is the p-adic field and if v is the infinite place then the completion is the real number field. Over the reals, (a, b)∞ is +1 if at least one of a or b is positive, and −1 if both are negative. Over the p-adics with p odd, writing and , where u and v are integers coprime to p, we have , where and the expression involves two Legendre symbols. Over the 2-adics, again writing and , where u and v are odd numbers, we have , where It is known that if v ranges over all places, (a, b)v is 1 for almost all places. Therefore, the following product formula makes sense. It is equivalent to the law of quadratic reciprocity. Kaplansky radical The Hilbert symbol on a field F defines a map where Br(F) is the Brauer group of F. The kernel of this mapping, the elements a such that (a,b)=1 for all b, is the Kaplansky radical of F. The radical is a subgroup of F*/F*2, identified with a subgroup of F*. The radical is equal to F* if and only if F has u-invariant at most 2. In the opposite direction, a field with radical F*2 is termed a Hilbert field. The general Hilbert symbol If K is a local field containing the group of nth roots of unity for some positive integer n prime to the characteristic of K, then the Hilbert symbol (,) is a function from K*×K* to μn. In terms of the Artin symbol it can be defined by Hilbert originally defined the Hilbert symbol before the Artin symbol was discovered, and his definition (for n prime) used the power residue symbol when K has residue characteristic coprime to n, and was rather complicated when K has residue characteristic dividing n. Properties The Hilbert symbol is (multiplicatively) bilinear: (ab,c) = (a,c)(b,c) (a,bc) = (a,b)(a,c) skew symmetric: (a,b) = (b,a)−1 nondegenerate: (a,b)=1 for all b if and only if a is in K*n It detects norms (hence the name norm residue symbol): (a,b)=1 if and only if a is a norm of an element in K() It has the "symbol" properties: (a,1–a)=1, (a,–a)=1. Hilbert's reciprocity law Hilbert's reciprocity law states that if a and b are in an